Ceres analytic derivation

1. Analytical derivative

In some cases , Don't use AutoDiffCostFunction, For example, we calculate the derivative in an approximate way , Rather than using AutoDiff The chain rule of , We need our own residuals and Jacobin Calculation . Now we define a CostFunction perhaps SizedCostFunction Subclasses of .

#include <vector>
#include "ceres/ceres.h"
#include "glog/logging.h"
 
using ceres::CostFunction;
using ceres::SizedCostFunction;
using ceres::Problem;
using ceres::Solver;
using ceres::Solve;
 
// A CostFunction implementing analytically derivatives for the
// function f(x) = 10 - x.
class QuadraticCostFunction : 
public SizedCostFunction<1 /* number of residuals */,
                         1 /* size of first parameter */> {
 public:
  virtual ~QuadraticCostFunction() {}
 
  virtual bool Evaluate(double const* const* parameters,
                        double* residuals,
                        double** jacobians) const {
    double x = parameters[0][0];
 
    // f(x) = 10 - x.
    residuals[0] = 10 - x;
 
    if (jacobians != NULL && jacobians[0] != NULL) {
      jacobians[0][0] = -1;
    }
 
    return true;
  }
};
 
int main(int argc, char** argv) {
  google::InitGoogleLogging(argv[0]);
 
  //  Initialize the variable to be optimized 
  double x = 0.5;
  const double initial_x = x;
 
  //  Construction issues 
  Problem problem;
 
  //  Set residual function 
  CostFunction* cost_function = new QuadraticCostFunction;
  problem.AddResidualBlock(cost_function, NULL, &x);
 
  //  Build solver 
  Solver::Options options;
  //  Output to cout
  options.minimizer_progress_to_stdout = true;
  //  Optimize information 
  Solver::Summary summary;
  //  Start calculating 
  Solve(options, &problem, &summary);
 
  std::cout << summary.BriefReport() << "\n";
  std::cout << "x : " << initial_x
            << " -> " << x << "\n";
 
  return 0;
}

The jacobians array is optional, Evaluate is expected to check when it is non-null, and if it is the case then fill it with the values of the derivative of the residual function.

QuadraticCostFunction::Evaluateparameters As input ,residuals Output as residual and Jacobians As the output of Jacobian matrix .jacobians It's optional ,Evaluate is expected to check when it is non-null, and if it is the case then fill it with the values of the derivative of the residual function. In this case , The residual function is linear , so Jacobian Is a constant .

Unless there is a good management Jacobian The reason of , Otherwise, it is recommended to use AutoDiffCostFunction or NumericDiffCostFunction.

2. CMakeLists.txt file

cmake_minimum_required(VERSION 2.8)
project(ceres)
find_package(Ceres REQUIRED)
include_directories(${CERES_INCLUDE_DIRS})
add_executable(use_ceres main.cpp)
 
target_link_libraries(use_ceres ${CERES_LIBRARIES})

More information about derivatives

up to now , The derivative is calculated using Ceres The most complex part of , And according to the situation , Users may need more complex methods of calculating derivatives . This section only describes how to provide derivatives to Ceres.

When you are familiar with NumericDiffCostFunction and AutoDiffCostFunction After the use of , It is suggested to consider DynamicAutoDiffCostFunction, CostFunctionToFunctor,NumericDiffFunctor as well as  ConditionedCostFunction More advanced methods to build and calculate cost functions .